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Keywords:
nonlinear string equation; accelerated convergence; existence; periodic; Dirichlet boundary conditions; vibrations; damped extensive string; time-periodic solution
nonlinear string equation; accelerated convergence; existence; periodic; Dirichlet boundary conditions; vibrations; damped extensive string; time-periodic solution
Summary:
In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.
References:
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[2] S. Klainerman: Global existence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), pp. 43--101. DOI 10.1002/cpa.3160330104 | MR 0544044 | Zbl 0405.35056
[3] P. Krejčí: Hard Implicit Function Theorem and Small periodic solutions to partial differential equations. Comment. Math. Univ. Carolinae 25 (1984), pp. 519-536. MR 0775567
[4] J. Moser: A rapidly-convergent iteration method and nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa 20-3 (1966), pp. 265-315, 499-535.
[5] O. Vejvoda, et al.: Partial differential equations: Time-periodic Solutions. Martinus Nijhoff Publ., 1982. Zbl 0501.35001
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